TITLE: What is the homotopy fiber of the map from a space to its James construction?
QUESTION [6 upvotes]: Given a pointed space X, let $J(X)$ denote its James construction. There is a natural inclusion $X\rightarrow J(X)$ which can equivalently be described as the unit map $\eta_X:X\rightarrow \Omega \Sigma X$.
The homotopy fiber of the counit map $\epsilon_X:\Sigma \Omega X\rightarrow X$ is known to be $\Omega X * \Omega X$. Is there an analogous result for the homotopy fiber of $\eta_X$ in terms of $X$?

REPLY [2 votes]: This might not be a satisfactory answer, but let me point out that S.-c. Wong constructed a "little cube" model for the homotopy fiber of $\Omega^{k-1}E^n : \Omega^{k-1}\Sigma^{k}X \to \Omega^{k-1+n}\Sigma^{k+n} X$ in this paper in 1994.
Note that May's little cube model for $\Omega\Sigma X$ is essentially the James construction.